Soak salt cod in a large pot of water in refrigerator for 36–48 hours, changing water at least 4 times.
Drain salt cod, then return to same pot, add water to barely cover, then bring to a boil. Reduce heat to low and poach salt cod for about 20 minutes, or until tender enough to pierce easily with a knife. Remove salt cod from pot and drain, reserving poaching liquid, then place fish in a large bowl to cool slightly. When it is cool enough to handle, pick through it carefully to remove any pieces of bone, cartilage, or skin.
Peel and quarter potatoes, then cook them over medium-high heat in reserved poaching water until tender, about 15 minutes. Drain, allow to cool slightly, then add to salt cod.
Wash and trim parsley, discarding stems, then dry and coarsely chop leaves. Add parsley and onion to potatoes and salt cod. Toss carefully (your hands work best), allowing potatoes and fish to break up slightly. Drizzle with olive oil, season with pepper, and toss again. Serve at room temperature or refrigerate until serving (salad will keep for several days if refrigerated).